Abstract
For first-order quasi-linear systems of partial differential equations, we formulate an assumption of a transition from initial hyperbolicity to ellipticity. This assumption bears on the principal symbol of the first-order operator. Under such an assumption, we prove a strong Hadamard instability for the associated Cauchy problem, namely an instantaneous defect of Hölder continuity of the flow from Gσ to L2, with 0<σ<σ0, the limiting Gevrey index σ0 depending on the nature of the transition. We restrict here to scalar transitions, and non-scalar transitions in which the boundary of the hyperbolic zone satisfies a flatness condition. As in our previous work for initially elliptic Cauchy problems [B. Morisse, On hyperbolicity and Gevrey well-posedness. Part one: the elliptic case, arXiv:1611.07225], the instability follows from a long-time Cauchy–Kovalevskaya construction for highly oscillating solutions. This extends recent work of N. Lerner, T. Nguyen, and B. Texier [The onset of instability in first-order systems, to appear in J. Eur. Math. Soc.].
Highlights
We consider the following Cauchy problem, for first-order quasi-linear systems of partial differential equations: d∂t u = Aj (t, x, u)∂xj u + f (t, x, u), j =1 u(0, x) = h(x). (1.1)The system is of size N, that is u(t, x) and f (t, x, u) are in RN and the Aj (t, x, u) ∈ RN×N
Under assumptions of weak defects of hyperbolicity for the first-order operator, we prove ill-posedness of (1.1) in Gevrey spaces
Our results extend Métivier’s ill-posedness theorem in Sobolev spaces for initially elliptic operators [10], our own ill-posedness result in Gevrey spaces for initially elliptic operators [11], Lerner, Nguyen and Texier’s theorem on systems transitioning from hyperbolicity to ellipticity [6], and echo Lu’s construction of WKB profiles [8] which are destabilized by terms not present in the initial data
Summary
We consider the following Cauchy problem, for first-order quasi-linear systems of partial differential equations: d. The time t is nonnegative, and x is in Rd. We assume throughout the paper that the Aj and f are analytic in a neighborhood of some point (0, x0, u0) ∈ Rt × Rdx × RNu. Under assumptions of weak defects of hyperbolicity for the first-order operator, we prove ill-posedness of (1.1) in Gevrey spaces. Our results extend Métivier’s ill-posedness theorem in Sobolev spaces for initially elliptic operators [10], our own ill-posedness result in Gevrey spaces for initially elliptic operators [11], Lerner, Nguyen and Texier’s theorem on systems transitioning from hyperbolicity to ellipticity [6], and echo Lu’s construction of WKB profiles [8] which are destabilized by terms not present in the initial data. Our assumptions of weak defects of hyperbolicity mean that the operator in (1.1) experiences a transition in time from hyperbolicity to non hyperbolicity. In the companion paper [12], we consider the case of genuinely non-scalar transitions
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